Interval Routing and Minor-Monotone GraphParameters

Erwin M. Bakker

Hans L. Bodlaender

Richard B. Tan

Jan van Leeuwen

Department of Information and Computing Sciences, Utrecht University

Technical Report UU-CS-2006-001

www.cs.uu.nl

ISSN: 0924-3275

Interval Routing and Minor-Monotone Graph Parameters∗

Erwin M. Bakker † Hans L. Bodlaender‡ Richard B. Tan§

Jan van Leeuwen¶

Abstract

We survey a number of minor-monotone graph parameters and their relationship to thecomplexity of routing on graphs. In particular we compare the interval routing parametersκslir(G) and κsir(G) with Colin de Verdiere’s graph invariant µ(G) and its variants λ(G)and κ(G). We show that for all the known characterizations of θ(G) with θ(G) being µ(G),λ(G) or κ(G), that θ(G) ≤ 2κslir(G) − 1 and θ(G) ≤ 2κsir(G) and conjecture that theseinequalities always hold. We show that θ(G) ≤ 4κslir(G) − 1 and θ(G) ≤ 4κsir(G) + 1.

1 Introduction

Graphs are a common model for the interconnection structure of communication networks.The complexity of the underlying graph should be an indication of the complexity of routinginformation in the network. In this paper we survey a number of minor-monotone graphparameters and their relationship to the complexity of routing using a particular type ofrouting schemes. We focus on Colin de Verdiere’s graph parameter µ(G) and its variantsλ(G) and κ(G) as a measure of complexity for the underlying graph, and on interval routingas the routing scheme of choice ([29, 30]).

Colin de Verdiere’s graph parameter µ(G) [9] is defined for any undirected graph G. It hasgenerated quite some interest due to its rather nice graph-theoretical properties. For example,µ(G) ≤ 3 if and only if G is planar, and µ(G) ≤ 4 if and only if G is linklessly embeddable inR

3 [25, 20]. Furthermore, µ(G) is minor-monotone, so by the graph minor theory of Robertsonand Seymour (see e.g., [24]) it can be characterized by means of forbidden minors. Van derHolst, Laurent and Schrijver [16, 14] introduced related minor-monotone parameters λ(G)and κ(G) based on extensions of µ(G). Characterization results for small values of the aboveparameters are known and surveyed in e.g. [18, 27].

∗This research was partially supported by project TACO (‘Treewidth and Combinatorial Optimization’)and by project BRICKS (‘Basic Research in Informatics for Creating the Knowledge Society’).

†LIACS, Leiden University, Niels Bohrweg 1, 2333 CA Leiden. erwin@liacs.nl‡Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB

Utrecht, the Netherlands. hansb@cs.uu.nl§Department of Information and Computing Sciences, Utrecht University, and Department of Computer

Science, University of Sciences & Arts of Oklahoma, Chickasha, OK 73018, USA. rbtan@cs.uu.nl¶Department of Information and Computing Sciences, Utrecht University. jan@cs.uu.nl

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Interval routing was first introduced by Santoro and Khatib [26], and subsequently generalizedby van Leeuwen and Tan [29, 30]. In interval routing schemes, the nodes of a graph arenumbered and the edges are labeled with intervals in order to allow the proper routing ofmessages to their set destinations. Interval routing schemes proved useful in the design oflarge processor networks and more recently, in routing issues for wireless ad-hoc networks.We are interested in the graph parameter κsir(G) and its variant κslir(G) that represent thecomplexity of a certain type of interval routing schemes.

The parameters κsir(G) and κslir(G) originate from the work by Frederickson and Janardan[12]. They introduced a variant of interval routing with ‘dynamic link costs’, where the cost ofthe edges in a graph can vary in such a way as to preserve the shortest path information. Forthe case of ‘strict’ interval routing schemes to be defined later, κsir(G) is the maximum numberof intervals minimally needed per edge to always be able to achieve shortest path routingby a suitable assignment of intervals. Frederickson and Janardan showed that κsir(G) =1 if and only if G is outerplanar. The relevance for this survey stems from the result ofBodlaender, Tan, Thilikos and van Leeuwen [8] that κsir(G) is minor-monotone and relatedto the treewidth of a graph, another well-known minor-monotone parameter due to Robertsonand Seymour. The parameter κslir(G) has similar properties and refers to a related, morerestricted lass of interval routing schemes. For surveys on interval routing, see [13, 28].

In this paper we compare the Colin de Verdiere parameter µ(G) and its variants λ(G) andκ(G), with the interval routing parameter κsir(G) and its variant κslir(G). We show that,with θ(G) denoting µ(G), λ(G) or κ(G):

i. κslir(G) = κsir(G) = 0 ⇐⇒ θ(G) = 0,

ii. κslir(G) = 1 =⇒ θ(G) ≤ 1,

iii. κsir(G) ≤ 1 =⇒ θ(G) ≤ 2,

iv. κslir(G) ≤ 2 =⇒ θ(G) ≤ 3,

v. κslir(G) ≤ bn2 c ⇐⇒ θ(G) ≤ n − 1 , where n is the size of G,

vi. κsir(G) ≤ dn2 e ⇐⇒ θ(G) ≤ n − 1.

For the invariants µ(G) and κ(G), we also have

i. κsir(G) ≤ 2 =⇒ µ(G) ≤ 4,

ii. 0 < κ(G) ≤ 2κslir(G) − 1 for all graphs G,

iii. κ(G) ≤ 2κsir(G) for all graphs G.

Thus, we know for some values of θ(G) the following relations and we conjecture that thesehold always:

i. 0 < θ(G) ≤ 2κslir(G) − 1 and

ii. θ(G) ≤ 2κsir(G).

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For general graphs G we can show the following bounds:

i. 0 < θ(G) ≤ 4κslir(G) − 1, and

ii. θ(G) ≤ 4κsir(G) + 1.

The paper is organized as follows. The next section gives some preliminaries on graph theory.In Section 3 we explain interval routing schemes and define the parameters κsir and κslir

associated with these. In Section 4 we define treewidth and relate it to the interval routingparameters. Section 5 deals with the Colin de Verdiere invariant µ(G) and its relationshipwith κsir(G) and κslir(G). Sections 6 and 7 contain definitions and results pertained to theparameters λ(G) and κ(G) respectively. The last section gives some conclusions and openproblems.

2 Preliminaries

Let G = (V,E) be a finite simple undirected graph. Let the set of vertices be V = {1, · · · , n}and write ij ∈ E if the edge {i, j} ∈ E.

We need the following graph operations. The definitions illustrate some of the manners inwhich a new graph can be created from component graphs.

Definition 1 (Clique-sum) A graph G = (V,E) is a clique-sum of graphs G1 = (V1, E1)and G2 = (V2, E2) if V = V1 ∪ V2, E = E1 ∪ E2 and V1 ∩ V2 is a clique in G1 and G2.

Writing G as a clique-sum is a decomposition technique. Often a parameter for G can bededuced from the parameters of G1 and G2. For example, for a graph G which is a clique-sumof graphs G1 and G2, the chromatic number χ(G) =max{χ(G1), χ(G2)}.

Definition 2 (∆Y -transformation) For a graph G,

i. the ∆Y -operation is as follows: choose a triangle in G and insert a new node v insidethe triangle and connect it to all the three nodes of the triangle, and then delete thetriangle.

ii. The Y ∆-operation is the reverse operation: starting with a node v of degree three, makeits three neighbors pairwise adjacent and then delete v with its three edges.

See Figure 3 for an example of ∆Y -transformation.

Definition 3 (Product graphs) For graphs G1 = (V1, E1) and G2 = (V2, E2), the productG = G1 × G2 is the graph with V = V1 × V2 and E = {{(u, v1), (u, v2)} : v1v2 ∈ E2}} ∪{{(u1, v), (u2, v)} : u1u2 ∈ E1}}, where u ∈ V1 and v ∈ V2.

Definition 4 A subdivision of a graph G is a graph obtained by inserting a new node alongan edge of G.

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Edge contraction is the operation of replacing an edge ij ∈ E by a new node v, making vadjacent to all nodes to which i and j are adjacent in G, and subsequently deleting nodes iand j and all their incident edges from G ([31]).

We now come to the important definition of a graph minor.

Definition 5 (Graph minors)

i. A graph H is a minor of graph G if H can be obtained from G by a series of zero ormore node deletions, edge deletions and edge contractions.

ii. A class of graphs G is minor-closed if for every G ∈ G, every minor H of G belongs toG.

iii. A proper minor of a graph G is a minor unequal to G.

iv. A graph H is a forbidden minor of the class of graphs G, if H 6∈ G but every properminor of H is.

v. A function θ : G → N is minor-monotone, if θ(H) ≤ θ(G) whenever G,H ∈ G and H isa minor of G.

Well-known classes of minor-closed graphs include the classes of: disjoint unions of paths,trees, outerplanar graphs, planar graphs, graphs that are knotlessly embeddable in R

3, andgraphs that are linklessly embeddable in R

3 (see [25]).

Fact 6 (Robertson and Seymour [24]) For every minor-closed class of graphs, the min-imal set of forbidden minors is finite.

It follows that every minor-closed class of graphs G has a finite characterisation in terms of afinite collection of forbidden minors F : G ∈ G if and only if G does not contain any graph ofF as a minor.

3 Interval Routing

Interval routing stems from compact routing methods in computer networks. Given a con-nected graph G with n nodes, an Interval Labeling Scheme (ILS) is a labeling of each nodev ∈ V with a unique name from {1, · · · , n} and of each incident edge (u, v) with a, possiblyempty, interval [a, b], where a, b ∈ {1, · · · , n}. If a > b then [a, b] = {a, a + 1, · · · , n, 1, · · · , b},i.e. wraparound is allowed. For each node u, with label `(u), the set of labels assigned toall incident edges forms a partition of the set {1, · · · , n}. In more general labelling schemes,more than one interval may be assigned to an incident edge.

Given an ILS, routing is done as follows. When a node receives a packet containing destinationaddress w, it checks its set of edge intervals (routing table) for the interval that contains wand relays the packet via the edge marked with that interval; unless w = `(v), in which casethe packet was routed correctly and is not relayed any further. It can be shown that every

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graph admits an ILS such that messages are always routed correctly to their destination nomatter where they depart, even when only one interval per incident edge is allowed and allintervals are required to be non-empty ([30]).

Normally each edge of G has a length, which is a non-negative integer. An ILS is valid if for allnodes v,w, all packets from v to w are routed correctly and eventually reach their destinationvia a path of minimum total length. An Interval Routing Scheme (IRS) is a valid ILS.

There are several variants of Interval Routing schemes, see [13, 28]. For our purpose, we needthe following:

Definition 7 (Strict interval routing schemes)

i. An IRS is Strict (SIRS) if for each node u, none of the intervals assigned to its incidentedges contains its own label, `(u).

ii. A k-SIRS is a SIRS where each incident edge is labeled with at most k intervals.

iii. κslir(G) is the smallest k, such that there exists a labeling ` of the nodes of V with uniquenames from {1, · · · , n}, with the property that for each assignment of non-negativelengths to the edges, there is a k-SIRS using labeling `.

iv. A Strict Linear Interval Routing Scheme (SLIRS) is a SIRS where the interval labelsare not allowed to wrap-around, i.e. for all non-empty interval labels [a, b], a ≤ b.

v. κslir(G) is the smallest k, such that there exists a labeling ` of the nodes of V with uniquenames from {1, · · · , n}, with the property that for each assignment of non-negativelengths to the edges, there is a k-SLIRS using labeling `.

The parameters κsir(G) and κslir(G) represent the routing complexity of a graph.

Fact 8 (Frederickson, Janardan [12])

i. κsir(G) = 1 ⇐⇒ G is outerplanar.

ii. K2,2k+1 and K2k+2 are forbidden minors for the class of graphs {G | κsir(G) = k}.

Fact 9 (Bodlaender, Tan, Thilikos, van Leeuwen [8]) The graph parameters κslir(G)and κsir(G) are minor-monotone.

The following bounds are obvious.

Fact 10 (Upper bounds)

i. κslir(G) ≤ bn2 c.

ii. κsir(G) ≤ dn2 e.

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Fact 11 (Bakker, Bodlaender, Tan, van Leeuwen [5])

i. Kp,q is a forbidden minor for the class of graphs {G | κslir(G) = k ≥ 1}, where 2 ≤ p ≤ qand p + q = 2k + 2.

ii. K2k+1 is a forbidden minor for the class of graphs {G | κslir(G) = k ≥ 1}.iii. Kp,q is a forbidden minor for the class of graphs {G | κsir(G) = k ≥ 1}, where 2 ≤ p ≤ q

and p + q = 2k + 3.

iv. κslir(G) = 1 ⇐⇒ G is a path.

4 Interval Routing and Treewidth

In this section we give the definition of treewidth of a graph and some results relating intervalrouting and treewidth.

Definition 12 A k-tree is a graph defined recursively as follows: A clique with k + 1 nodesis a k-tree. A k-tree with n + 1 nodes can be formed from a k-tree with n nodes by adding anew node and making it adjacent to exactly all nodes of a k-clique in the original k-tree.

Definition 13 A graph is a partial k-tree if it is a subgraph of a k-tree.

Definition 14 The treewidth τω(G) of a graph G is the minimum value k for which thegraph is a subgraph of a k-tree.

The definition above is due to Arnborg and Proskurowski, see e.g. [1]. The term treewidth andthe commonly used definition in terms of tree decompositions was introduced by Robertsonand Seymour [22]. There are other well-known equivalent definitions, see the surveys byBodlaender [6, 7], for instance.

Fact 15 (Treewidth)

i. τω(G) = 1 ⇐⇒ G is a tree.

ii. τω(G) ≤ 2 ⇐⇒ G does not contain K4 as a minor.

iii. (Arnborg, Proskurowski, Corneil [2]) τω(G) ≤ 3 ⇐⇒ G does not have a minorisomorphic to any of the graphs in Figure 1.

iv. τω(Kn) = n − 1.

Colin de Verdiere [11] defines a variant of the notion of treewidth which is simpler.

Definition 16 τωCdV (G) (denoted la(G) in [11]) is the smallest integer n such that G is aminor of T × Kn, where T is a tree.

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Figure 1: Forbidden Minors of τω(G) ≤ 3

The two definitions of treewidth are very closely related and in fact their values differ by atmost 1.

Fact 17 (Colin de Verdiere [11], van der Holst [14])τω(G) ≤ τωCdV (G) ≤ τω(G) + 1.

Fact 18 (Bodlaender,Tan,Thilikos,van Leeuwen [8])

i. τω(G) ≤ 4κslir(G) − 2.

ii. τω(G) ≤ 4κsir(G).

Corollary 19

i. τωCdV (G) ≤ 4κslir(G) − 1.

ii. τωCdV (G) ≤ 4κsir(G) + 1.

For known values of treewidth, the bounds can be improved.

Theorem 20 Let G be a connected graph.

i. κslir(G) = 1 =⇒ τω(G) = 1.

ii. κsir(G) = 1 =⇒ τω(G) ≤ 2.

iii. κslir(G) ≤ 2 =⇒ τω(G) ≤ 3.

iv. κsir(G) ≤ dn2 e ⇐⇒ τω(G) ≤ n − 1.

v. κslir(G) ≤ bn2 c ⇐⇒ τω(G) ≤ n − 1.

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Proof. (i) κslir(G) = 1 ⇐⇒ G is a path, so the equality holds.(ii) κsir(G) = 1 ⇐⇒ G is an outerplanar graph, and outerplanar graphs cannot contain K4

or K2,3 as a minor, so have treewidth at most 2.(iii) See Figure 1. Each graph from the set of forbidden minors for the class of graphs withτω(G) ≤ 3 contains K5 (the first graph), K2,4 (the next two graphs) or K3,3 (the last graph);all are also forbidden minors for the class of graphs with κslir(G) = 2. So when κslir(G) ≤ 2,τω(G) cannot be larger than 3. The bound is tight as κslir(K4) = 2 and τω(K4) = 3.(iv)(v) Follows from the upper bounds when G = Kn. ut

Thus we can sharpen the bounds to τω(G) ≤ 2κslir(G) − 1 and τω(G) ≤ 2κsir(G) for knowncharacterizations of τω(G).

5 Interval Routing and µ(G)

In this section we give the relations between the interval routing parameters κslir(G) andκsir(G) with the Colin de Verdiere’s invariant µ(G) introduced in [9]. The motivation of thedefinition stems from Schrodinger operators in differential geometry and is rather technicalin nature. We introduce three conditions on matrices, related to graphs.

Condition (M1)(Discrete Schrodinger Operator)OG is the set of real symmetric V × V matrices M = (Mij) such that

i. Mij < 0 if ij ∈ E, and

ii. Mij = 0 if i 6= j and ij 6∈ E.

There is no restriction on the diagonal entries of M .

As M ∈ OG is real symmetric, M has n real eigenvalues (counting multiplicities) λ1 ≤ λ2 ≤· · · ≤ λn, called the spectrum of M . If G is connected, then by Perron-Frobenius Theoremthe first eigenvalue λ1 must be of multiplicity 1. There is no loss of generality in restrictingattention to connected graphs, since the spectrum of a disconnected graph is the union of thespectra of its connected components.

Condition (M2)(Normalization)M ∈ OG has exactly one negative eigenvalue, of multiplicity 1.

As there is no restriction on the diagonal entries, we can replace M by M − λ2I. Then thenew λ1 is the only negative eigenvalue and the new λ2 now is zero. Thus by shifting thespectrum of M this condition can always be achieved.

Condition (M3)(Strong Arnol’d Hypothesis - SAH)There is no nonzero real symmetric matrix X such that MX = 0 with Xij = 0 wheneveri = j or ij ∈ E.

This is a non-degeneracy condition on M . It is equivalent to the concept of transversalintersection of differential manifolds. Transversal intersections have nice properties because

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near them the manifolds behave like affine subspaces. (See [18], pp. 8-9, for other equivalentconditions of SAH.)

Definition 21 A matrix M satisfying conditions (M1), (M2) and (M3) is called a Colin deVerdiere matrix for G.

We are now ready to define the Colin de Verdiere’s invariant.

Definition 22 µ(G) is the maximum corank of any Colin de Verdiere matrix of G.

As noted above, the second eigenvalue λ2 can be shifted such that it is zero, with the samemultiplicity. Thus Mx = λ2x = 0x = 0 and the dimension of the kernel (null space) of M , isthe corank of M . This coincides with the multiplicity of λ2. So µ(G) can also be defined asthe maximum multiplicity of the second smallest eigenvalue of M satisfying conditions (M1)and (M3), i.e. M ∈ OG satisfying SAH. For more detailed explanations of µ(G), see Colin deVerdiere’s original paper [9] ([10] for an English translation) or [18] for a more leisurely pacedsurvey.

We now state some known results of µ(G).

Fact 23 (Colin de Verdiere [9, 11])

i. µ(G) is minor-monotone.

ii. µ(G) = 0 ⇐⇒ G is a single node.

iii. µ(G) = 1 ⇐⇒ G is a disjoint union of paths.

iv. µ(G) ≤ 2 ⇐⇒ G is outerplanar.

v. µ(G) ≤ 3 ⇐⇒ G is planar.

vi. µ(G) ≤ n − 1.

vii. µ(Kn) = n − 1.

viii. µ(G) ≤ τωCdV (G) ≤ τω(G) + 1.

A graph G is linklessly embeddable if it can be embedded in R3 so that any two disjoint circuits

in G form unlinked closed curves in R3. (See [25] for a more detailed explanation.) The

following results are due to Robertson, Seymour and Thomas[25] and Lovasz and Schrijver[20].

Fact 24 (Linkless embedding)

i. (Robertson, Seymour and Thomas [25]) µ(G) ≤ 4 =⇒ G is linklessly embeddable.

ii. (Lovasz and Schrijver [20]) G is linklessly embeddable =⇒ µ(G) ≤ 4.

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We need the following characterization of a linklessly embeddable graph.

Definition 25 ((Petersen Family)) The Petersen Family consists of the seven graphs inFigure 2. The first one is K6 and the others are its ∆Y -transformations, one of which is thePetersen graph.

Fact 26 (Robertson, Seymour and Thomas[25]) A graph G is linklessly embeddable⇐⇒ G has no minor isomorphic to a member of the Petersen family.

Figure 2: The Petersen Family

Theorem 27 Let G be a connected graph.

i. κslir(G) = κsir(G) = 0 ⇐⇒ µ(G) = 0.

ii. κslir(G) = 1 ⇐⇒ µ(G) = 1.

iii. κsir(G) = 1 ⇐⇒ µ(G) ≤ 2.

iv. κslir(G) = 2 =⇒ µ(G) ≤ 3.

v. κsir(G) = 2 =⇒ µ(G) ≤ 4.

vi. κslir(G) ≤ bn2 c ⇐⇒ µ(G) ≤ n − 1.

vii. κsir(G) ≤ dn2 e ⇐⇒ µ(G) ≤ n − 1.

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Proof.

i. This is trivially true.

ii. κslir(G) = 1 implies that G is a path, so µ(G) = 1.

iii. This follows from the fact that κsir(G) = 1 if and only if G is outerplanar if and only ifµ(G) ≤ 2.

iv. If κslir(G) = 2, then G does not contain K5 or K3,3 as a minor, so G is planar. Com-bining this with Colin de Verdiere’s result of µ(G) ≤ 3 yields the inequality. The boundis tight as κslir(K4) = 3.

v. Suppose there is a G with κsir(G) = 2 but µ(G) > 4. Then G must contain a minor inthe Petersen family. Now, K6 and K3,4 are both forbidden minors for G. By inspection,each graph from the Petersen family contains both of these as minors: a contradiction.The bound is tight as κsir(K5) = 2 and µ(K5) = 4.

vi. Follows from the upper bounds when G = Kn.

vii. Again follows from the case G = Kn.

ut

The above theorem shows that for several possible values of κslir(G) and κsir(G), we haveµ(G) ≤ 2κslir(G)−1 and µ(G) ≤ 2κsir(G). It is tempting to generalize this to any G. We for-mulate it as a conjecture based on the evidence that results from the known characterizationsto date.

Conjecture 28 For any G and µ(G) > 0,

i. µ(G) ≤ 2κslir(G) − 1.

ii. µ(G) ≤ 2κsir(G).

We can prove the following, weaker result.

Theorem 29 For κslir(G),κsir(G) ≥ 3,

i. µ(G) ≤ 4κslir(G) − 1.

ii. µ(G) ≤ 4κsir(G) + 1.

Proof. Since µ(G) ≤ τω(G) + 1 and τω(G) ≤ 4κslir(G) − 2, τω(G) ≤ 4κsir(G), theinequalities follow. ut

We now consider some of the effects on µ(G), κslir(G) and κsir(G) of some common graphoperations.

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Fact 30 (Colin de Verdiere [9]) µ(G) ≤ µ(G − v) + 1

In essence, the result states that when a node v and its associated edges are deleted from G,µ(G) and µ(G− v) differs by at most one. We show that this is not the case for κslir(G) andκsir(G), except for the simple cases when κslir(G) ≤ 1. In fact, κslir(G) and κslir(G− v) candiffer by as much as we want (up to κslir(G)).

Theorem 31 For any constant s ≥ 0,

i. there exists a graph G and a node v such that κslir(G) ≥ κslir(G − v) + s,

ii. there exists a graph G and a node v such that κsir(G) ≥ κsir(G − v) + s.

Proof. (i) When κslir(G) = 1, let G = K1,2. If s = 1, let node v be the interior node,then κslir(G) = 1 and κslir(G − v) = 0. If s = 0, let node v to be one of the leaf nodes, thenκslir(G − v) = 1. Thus the inequality holds.

Suppose now that κslir(G) ≥ 2. If s = 0, then κslir(G) = κslir(G + v), where G + v is G withan extra node attached to any node in G. Deleting this v will satisfy the inequality. Let s ≥ 1.Consider the graph G = K2,2s+2 and let v be one of the 2−nodes. Then G − v = K1,2s+2.Now, κslir(G) = s + 2 but κslir(G − v) ≤ 2, so κslir(G) ≥ κslir(G − v) + s.

(ii) The proof is similar as above, except that we take G = K2,2s+1. ut

We now investigate the clique-sum of graphs. In [17], it is shown that under the right condi-tions, µ(G) = max{µ(G1), µ(G2)}.

Theorem 32 For any constant s ≥ 0,

i. there is a graph G with clique-sum G1 and G2 such that κslir(G) ≥max{κslir(G1), κslir(G2)} + s;

ii. there is a graph G with clique-sum G1 and G2 such that κsir(G) ≥max{κsir(G1), κsir(G2)} + s.

Proof. When s = 0, we can just choose G = K1,2 with G1 = K2 = G2 intersecting ata single node. Then κslir(G) = κslir(G1) = κslir(G2) = 1. When s = 1, we let G1 = K1,2

and G2 = K2 with G = K1,3 so that G1 and G2 intersect at the interior node of G1. Thenκslir(G1) = κslir(G2) = 1 but κslir(G) = 2.

Assume now s ≥ 2. Let K+2,r denote the graph K2,r with an extra edge connecting the 2-

nodes. Let G1 = K+2,2s−1 = G2, G = G1 ∪G2 = K+

2,4s−2 and G1 ∩G2 = K2. Then κslir(G1) =κslir(G2) = s, but κslir(G) = κslir(K+

2,4s−2) = 2s. Thus κslir(G) ≥ max{µ(G1), µ(G2)} + s.

(ii) Similar to above. Choose G1 = K2,2s = G2. ut

Subdivision of G does not seem to affect µ(G) very much.

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Fact 33 (Colin de Verdiere [9]) Let G′ be a subdivision of G. Then

i. µ(G) ≤ µ(G′).

ii. µ(G) = µ(G′) if µ(G) ≥ 3.

Contrary to µ(G), κslir(G′) can be greater than κslir(G) for κslir(G) ≥ 2.

Theorem 34 (Subdivision)

i. For k ≥ 2, there exists a graph G and a subdivision G′ with k = κslir(G) < κslir(G′).

ii. For k ≥ 1, there exists a graph G and a subdivision G′ with k = κsir(G) < κsir(G′).

Proof. (i) Let G = K+2,2k−1. Then κslir(G) = k, but a subdivision along the edge connecting

the 2-nodes yields G′ = K2,2k, a forbidden minor.

(ii) Similar for κsir(G). Let G = K+2,2k. ut

However the values of κslir(G) and κsir(G′) can only differ by at most one.

Theorem 35 Let G′ be a subdivision of a graph G. Then

i. κslir(G′) ≤ κslir(G) + 1,

ii. κsir(G′) ≤ κsir(G) + 1.

Proof. Assume the edge where the subdivision occurs is adjacent to nodes u and u′ andthe new node is v. Then label node v with n+1. Any path to v must either go through nodeu or through u′, so in the worst case at each edge, we add the extra interval [v, v], i.e., wehave an increase with at most one extra interval. The edge (v, u) is labeled with the originalintervals of edge (u′, u) and likewise edge (v, u′) with original intervals of edge (u, u′), withno increase in the number of intervals. ut

Bacher and Colin de Verdiere [3] showed that for µ(G) ≥ 4, µ(G) is invariant under the ∆Y -and Y ∆-operations. We show that this is not the case for κslir(G) and κsir(G).

Theorem 36 (∆Y -transformations)

i. For k ≥ 2, there exists a graph G and a graph G′ obtained from G by a ∆Y -transformation, such that κslir(G) = k < κslir(G′).

ii. For k ≥ 1, there exists a graph G and a graph G′ obtained from G by a ∆Y -transformation, such that κsir(G) = k < κsir(G′).

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Proof. (i) Assume k ≥ 2. In the top of Figure 3 the graph ∆ which is outerplanar andcontains K2,2 is transformed to the graph Y which contains K2,3. By adding 2k − 3 extranodes and then connecting them to the graphs ∆ and Y , we obtain the graphs G and G′; eachof the extra nodes is made adjacent to two vertices, as is shown in Figure 3. Now κslir(G) = kand κslir(G′) = k + 1 as it contains the forbidden minor K2,2k.

(ii) Similar to the above construction, except that G is the ∆-graph with 2k − 2 extra nodesconnected to ∆. ut

G′G

Y∆

Figure 3: The ∆Y Transformation

It is an interesting open question what the maximum difference can be between κsir(G)and κsir(G′) and between κsir(G) and κsir(G′) when G′ is obtained from G by a ∆Y -transformation.

6 Interval Routing and λ(G)

In [16] van der Holst, Laurent and Schrijver introduced an interesting graph parameter basedon Colin de Verdiere’s µ(G).

Definition 37 (λ(G))

i. For any vector x ∈ Rn, the positive support is supp+(x) = {i|xi > 0}.

ii. A linear subspace L ⊂ Rn is a valid representation of G = (V,E) with V = {1, 2, · · · , n},

if for each nonzero x ∈ L, supp+(x) 6= ∅ and the subgraph induced by the set of nodessupp+(x) in G is connected.

iii. λ(G) = max{dim(L)|L is a valid representation of G}.

The motivation from µ(G) is that, when x is in the null space of a Colin de Verdiere matrixM (with minimal support), the subspace induced by supp+(x) is nonempty and connected.

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Fact 38 (van der Holst, Laurent, Schrijver [16])

i. λ(G) = 0 ⇐⇒ G has no K2 as a minor.

ii. λ(G) = 1 ⇐⇒ G has no K3 as a minor.

iii. λ(G) ≤ 2 ⇐⇒ G has no K4 as a minor.

iv. λ(G) ≤ 3 ⇐⇒ G has no K5 or V8 as a minor (V8 is the last graph in Figure 1).

v. λ(Kn) = n − 1.

vi. λ(G) is minor-monotone.

There is an interesting connection between µ(G) and λ(G).

Fact 39 (Pendavingh [21]) µ(G) ≤ λ(G) + 2.

It is conjectured that µ(G) ≤ λ(G) + 1.

We now consider the relationship between the interval routing parameters and λ(G).

Theorem 40 Let G be a connected graph.

i. κslir(G) = κsir(G) = 0 ⇐⇒ λ(G) = 0.

ii. κslir(G) = 1 =⇒ λ(G) = 1.

iii. κsir(G) = 1 =⇒ λ(G) ≤ 2.

iv. κslir(G) = 2 =⇒ λ(G) ≤ 3.

v. κslir(G) ≤ bn2 c ⇐⇒ λ(G) ≤ n − 1.

vi. κsir(G) ≤ dn2 e ⇐⇒ λ(G) ≤ n − 1.

Proof. The results follow from the forbidden minors of the parameters. For the inequality(iii), we note that the graph V8 contains K3,3 as a minor, which is forbidden for G withκslir(G) = 2. Again, the last two inequalities follow from the upper bounds of G = Kn. ut

The above theorem shows again that for several values of κslir(G) and κsir(G), λ(G) ≤2κslir(G) − 1 and λ(G) ≤ 2κsir(G). The following conjecture may also hold.

Conjecture 41 For any λ(G) > 0,

i. λ(G) ≤ 2κslir(G) − 1.

ii. λ(G) ≤ 2κsir(G).

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7 Interval Routing and κ(G)

In [16], van der Holst, Laurent and Schrijver introduced yet another graph parameter relatedto λ(G), which they called κ(G). The definition is rather technical but similar to λ(G).

Definition 42 (κ(G)) Let G = (V,E) be a connected graph and φ : V → Rd be a function

such that:

i. φ(V ) affinely spans a d-dimensional affine space,

ii. for each affine halfspace H ⊆ Rd, φ−1(H) induces a connected subgraph of G (possibly

empty).

κ(G) is the largest d for which the above conditions hold, for a suitable φ.

It turns out that κ(G) has a complete forbidden minor characterization.

Fact 43 (van der Holst, Laurent, Schrijver [16])

i. κ(G) is minor-monotone.

ii. κ(G) ≤ k ⇐⇒ G does not contain Kk+2 as a minor.

iii. κ(G) ≤ λ(G).

It turns out that Conjecture 41 actually can be shown for the case of κ(G).

Theorem 44 For every kappa ≥ 1,

i. κ(G) ≤ 2κslir(G) − 1.

ii. κ(G) ≤ 2κsir(G).

Proof. (i) This follows from the fact that K2k+1 is a forbidden minor of the class of graphsG with κslir(G) = k and κslir(K2k) = k, κ(K2k) = 2k − 1.

(ii) This also follows from the fact that K2k+2 is a forbidden minor of the class of graphs Gwith κsir(G) = k with κslir(K2k+1) = k and κ(K2k+1) = 2k. ut

8 Conclusion

In this paper, we have surveyed and explored several graph parameters that are indicativefor the complexity of a graph and for routing information in a graph. In particular we havemade comparisons between Colin de Verdiere’s graph parameter µ(G) and its variants λ(G)and κ(G), and the parameters κslir(G) and κsir(G) related to certain types of interval routing

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schemes. Interesting relationships could be observed based on the known characterisations ofthe parameters, which are all minor-monotone. With θ(G) denoting µ(G), λ(G) or κ(G), itappears that 0 < θ(G) ≤ 2κslir(G) − 1 and θ(G) ≤ 2κsir(G), for the known characterizationsof the respective parameters (and the case κ(G)).

It may be that the above inequalities also hold for any graph G but we leave this as aconjecture. Also, all the results derived are based on the known characterizations by forbiddenminors. It would be interesting to see if there are any direct proofs by using the definitions.

References

[1] S. Arnborg, A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J.Disc. Appl. Meth. 8 (1987) 277-284.

[2] S. Arnborg, A. Proskurowski, D.G. Corneil, Minimal forbidden minor characterizationof a class of graphs, Discrete Mathematics 80 (1990) 1-19.

[3] R. Bacher, Y. Colin de Verdiere, Multiplicites des valeurs propres et transformationsetoile- triangle des graphes, Bulletin de la Socieete Mathematique de France 123 (1995)101-117.

[4] E.M. Bakker, R.B. Tan, J. van Leeuwen, Linear interval routing schemes, AlgorithmsReview 2 (1991), 45-61.

[5] E.M. Bakker, H.L. Bodlaender, R.B. Tan, J. van Leeuwen, Interval routing schemes viagraph reductions, manuscript (2004).

[6] H.L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth, TheoreticalComputer Science 209 (1998), 1-45.

[7] H.L. Bodlaender, Discovering treewidth, in: SOFSEM 2005: Theory and Practice ofComputer Science, Proc. 31st Conference on Current Trends in Theory and Practice ofComputer Science (P. Vojtas et al., eds.), Lecture Notes in Computer Science, Vol. 3381,Springer-Verlag, Berlin, 2005, pp. 1-16.

[8] H.L. Bodlaender, R.B. Tan, D. Thilikos, J. van Leeuwen, On interval routing schemesand tree-width, Information and Computation 139 (1) (1997), 92-109.

[9] Y. Colin de Verdiere, Sur un nouvel invariant des graphes et un critere de planarite,Journal of Combinatorial Theory, Series B 50 (1990), 11-21.

[10] Y. Colin de Verdiere, On a new graph invariant and a criterion for planarity, in: GraphStructure Theory (N. Robertson and P. Seymour, eds.), Contemporary Mathematics,American Mathematical Society, Providence, Rhode Island (1993), 137-147.

[11] Y. Colin de Verdiere, Multiplicities of eigenvalues and tree-width of graphs, Journal ofCombinatorial Theory, Series B 74 (1998), 121-146.

[12] G.N. Frederickson, R. Janardan, Designing Networks with Compact Routing Tables,Algorithmica 3 (1988), 171-190.

17

[13] C. Gavoille, A survey on interval routing, Theoretical Computer Science, 245 (2) (2000),217-253.

[14] H. van der Holst, Topological and spectral graph characterizations, PhD thesis, Univer-sity of Amsterdam (1996).

[15] H. van der Holst, Graphs with magnetic Schrodinger operators of low corank, Journal ofCombinatorial Theory, Series B 84 (2002), 311-339.

[16] H. van der Holst, M. Laurent, A. Schrijver, On a minor-monotone graph invariant, Jour-nal of Combinatorial Theory, Series B 65 (1995), 291-304.

[17] H. van der Holst, L. Lovasz, A. Schrijver, On the invariance of Colin de Verdiere’s graphparameter under clique sums, Linear Algebra and Its Applications 226 (1995), 291-304.

[18] H. van der Holst, L. Lovasz, A. Schrijver, The Colin de Verdiere graph parameter, in:Graph Theory and Combinatorial Biology, Bolyai Soc. Math. Stud. 7, Janos Bolyai Math.Soc., Budapest (1999), 29-85.

[19] A. Kotlov, Spectral characterization of tree-width-two graphs, Combinatorica 20 (1)(2000), 147-152.

[20] L. Lovasz, A. Schrijver, A Borsuk theorem for antipodal links and a spectral charac-terization of linklessly embeddable graphs, Proceedings of the American MathematicalSociety, 126 (1998), 1275-1285.

[21] R. Pendavingh, On the relation between two minor-monotone graph parameters, Com-binatorica 18 (2) (1998), 281-292.

[22] N. Robertson, P.D. Seymour, Graph minors: II. Algorithmic aspects of tree-width, J.Algorithms7 (1986), 309-322.

[23] N. Robertson, P.D. Seymour, Some new results on the well-quasi ordering of graphs,Ann. Discr. Math. 23 (1987), 343-354.

[24] N. Robertson, P.D. Seymour, Graph minors: XX. Wagner’s conjecture, Journal Comb.Theory Ser. B 92 (2004), 325-357.

[25] N. Robertson, P.D. Seymour, R. Thomas, A survey of linkless embeddings, in: GraphStructure Theory (N. Robertson and P. Seymour, eds.), Contemporary Mathematics,American Mathematical Society, Providence, Rhode Island (1993), 125-136.

[26] N. Santoro, R. Khatib, Labelling and implicit routing in networks, The Computer Journal28 (1985), 5-8.

[27] A. Schrijver, Minor-monotone graph invariants, in: Surveys in Combinatorics (R.A.Bailey, ed.), Cambridge University Press, Cambridge (1997), 163-196.

[28] R.B. Tan, J. van Leeuwen, Compact routing methods on communication networks: Asurvey, in: Proceedings of the Colloquium on Structural Information and CommunicationComplexity (SIROCCO), Carleton University Press (1995), 99-110.

18

[29] J. van Leeuwen, R.B. Tan, Compact networks with compact routing tables, in: The Bookof L (G. Rozenberg and A. Salomaa, eds.), Springer-Verlag, Berlin (1986), 259-273.

[30] J. van Leeuwen, R.B. Tan, Interval routing, The Computer Journal 30 (1987), 298-307.

[31] Th. Wolle, H.L. Bodlaender, A note of edge contraction, Technical Report UU-CS-2004-028, Dept of Information and Computing Sciences, Utrecht University, Utrecht, 2004.

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